Average Error: 3.6 → 1.2
Time: 6.4m
Precision: 64
Internal Precision: 320
\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.1169717498132977 \cdot 10^{+167}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{\left(\beta + 1.0\right) + (\beta \cdot \alpha + \alpha)_*}{\left(\beta + 2\right) + \alpha}}{\left(\beta + 2\right) + \alpha}}}{\left(1.0 + \alpha\right) + \left(\beta + 2\right)} \cdot \sqrt{\frac{\frac{\left(1.0 + \alpha\right) + (\beta \cdot \alpha + \beta)_*}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) - \frac{1}{{\alpha}^{2}}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 4.1169717498132977e+167

    1. Initial program 1.3

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity1.3

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right)}}\]

    4. Applied add-sqr-sqrt1.4

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0\right)}\]

    5. Applied times-frac1.4

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{1} \cdot \frac{\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}}\]

    6. Simplified1.4

      \[\leadsto \color{blue}{\sqrt{\frac{\frac{(\beta \cdot \alpha + \beta)_* + \left(1.0 + \alpha\right)}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 2\right)}}} \cdot \frac{\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    7. Simplified1.4

      \[\leadsto \sqrt{\frac{\frac{(\beta \cdot \alpha + \beta)_* + \left(1.0 + \alpha\right)}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 2\right)}} \cdot \color{blue}{\frac{\sqrt{\frac{\frac{\left(\beta + 1.0\right) + (\beta \cdot \alpha + \alpha)_*}{\left(2 + \beta\right) + \alpha}}{\left(2 + \beta\right) + \alpha}}}{\left(1.0 + \alpha\right) + \left(2 + \beta\right)}}\]

    if 4.1169717498132977e+167 < alpha

    1. Initial program 16.3

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    2. Using strategy rm

    3. Applied clear-num16.4

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt16.4

      \[\leadsto \frac{\frac{\frac{1}{\frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    6. Applied *-un-lft-identity16.4

      \[\leadsto \frac{\frac{\frac{1}{\frac{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot 1\right)}}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    7. Applied times-frac16.4

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}} \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    8. Simplified16.4

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\frac{1}{\sqrt{\left(\alpha + 1.0\right) + (\beta \cdot \alpha + \beta)_*}}} \cdot \frac{\left(\alpha + \beta\right) + 2 \cdot 1}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    9. Simplified16.4

      \[\leadsto \frac{\frac{\frac{1}{\frac{1}{\sqrt{\left(\alpha + 1.0\right) + (\beta \cdot \alpha + \beta)_*}} \cdot \color{blue}{\frac{2 + \left(\beta + \alpha\right)}{\sqrt{\left(\alpha + 1.0\right) + (\beta \cdot \alpha + \beta)_*}}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

    10. Taylor expanded around inf 0.1

      \[\leadsto \frac{\frac{\frac{1}{\color{blue}{\left(\frac{1}{\alpha} + \frac{1}{\beta}\right) - \frac{1}{{\alpha}^{2}}}}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.1169717498132977 \cdot 10^{+167}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{\left(\beta + 1.0\right) + (\beta \cdot \alpha + \alpha)_*}{\left(\beta + 2\right) + \alpha}}{\left(\beta + 2\right) + \alpha}}}{\left(1.0 + \alpha\right) + \left(\beta + 2\right)} \cdot \sqrt{\frac{\frac{\left(1.0 + \alpha\right) + (\beta \cdot \alpha + \beta)_*}{\beta + \left(\alpha + 2\right)}}{\beta + \left(\alpha + 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) - \frac{1}{{\alpha}^{2}}}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0}\\ \end{array}\]

Runtime

Time bar (total: 6.4m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes3.71.20.03.667.2%
herbie shell --seed 2018297 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))